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Expenditure Minimization Probl

(2004-11-19 06:24:32) 下一個

Expenditure Minimization Problem

From Wikipedia, the free encyclopedia.

In microeconomics, the Expenditure Minimization Problem is the dual problem to the Utility Maximization Problem: how much money do I need to be happy?. This question comes in two parts. Given a consumer's utility function, prices, and a utility target,

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Expenditure function

Formally, the expenditure function is defined as follows. Suppose the consumer has a utility function u defined on L commodities. Then the consumer's expenditure function gives the amount of money required to buy a package of commodities at given prices p that give utility greater than u * ,

e(p, u^*) = \min_{x \in \geq(u^*)} p \cdot x

where

\geq(u^*) = \{x \in \textbf R^L_+ : u(x) \geq u^*\}

is the set of all packages that give utility at least as good as u * .

Hicksian demand correspondence

Secondly, the Hicksian demand correspondence h(p,u * ) is defined as the cheapest package that gives the desired utility. It can be defined in terms of the expenditure function with the Marshallian demand correspondence

h(p,u * ) = x(p,e(p,u * )).

If the Marshallian demand correspondence x(p,w) is a function (i.e. always gives a unique answer), then h(p,u * ) is also called the Hicksian demand function.

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